Avoiding (m, m, m)-Arrays of Order n=2k

An (m,m,m)-array of order n is an n×n array such that each cell is assigned a set of at most m symbols from {1, . . . , n} such that no symbol occurs more than m times in any row or column. An (m,m,m)-array is called avoidable if there exists a Latin square such that no cell in the Latin square contains a symbol that also belongs to the set assigned to the corresponding cell in the array. We show that there is a constant γ such that if m ≤ γ2k and k ≥ 14, then any (m,m,m)-array of order n = 2k is avoidable. Such a constant γ has been conjectured to exist for all n by Häggkvist.